Interpretation (model theory)

For other uses, see Interpretation (disambiguation).

In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the vague idea that M is (model-theoretically) at most as complicated as N. For example every reduct or definitional expansion of a structure N has an interpretation in N.

Many model-theoretic properties are preserved under interpretability. For example if the theory of N is stable and M is interpretable in N, then the theory of M is also stable.

Definition

An interpretation of M in N with parameters is a pair (n,f) where n is a natural number and f is a surjective map from a subset of Nⁿ onto M such that the f-preimage (more precisely the f^k-preimage) of every set X ⊆ M^k definable in M by a first-order formula without parameters is definable (in N) by a first-order formula with (possibly) parameters. An interpretation (n,f) with parameters is called an interpretation without parameters if the f-preimage of every set definable without parameters is also definable without parameters. Since the value of n for an interpretation (n,f) is often clear from the context, the map f itself is also called an interpretation.

It is customary in model theory to use the terms definable, 0-definable, interpretation, 0-interpretation instead of, respectively, definable with parameters, definable without parameters, interpretation with parameters, and interpretation without parameters.

If L, M and N are three structures, L is interpreted in M, and M is interpreted in N, then one can naturally construct a composite interpretation of L in N. If two structures M and N are interpreted in each other, then by combining the interpretations in two possible ways, one obtains an interpretation of each of the two structure in itself. This observation permits to define an equivalence relation among structures, reminiscent of the homotopy equivalence among topological spaces.

Two structures M and N are bi-interpretable if there exist an interpretation of M in N and an interpretation of N in M such that the composite interpretations of M in itself and of N in itself are definable in M and in N, respectively (the composite interpretations being viewed as operations on M and on N).

Example

The partial map $f$ of $\mathbb {Z} \times \mathbb {Z}$ onto $\mathbb {Q}$ defined by $(x,y)\mapsto {\frac {x}{y}}$ provides an interpretation of the field of rational numbers in the ring of integers (to be precise, the interpretation is $(2,f)$). In fact, this particular interpretation is often used to define the rational numbers.

References

• Ahlbrandt, Gisela; Ziegler, Martin (1986), "Quasi finitely axiomatizable totally categorical theories", Annals of Pure and Applied Logic, 30: 63–82
• Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6 (Section 4.3)
• Poizat, Bruno (2000). A Course in Model Theory. Springer. ISBN 0-387-98655-3. (Section 9.4)